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Higher-Order Action Supervision Makes A Strong Policy Class

Peng Cheng, Yunxian Hou, Zhi Zhou, Qian Zhang, Chang Huang, Xianyuan Zhan

arXiv:2610.11175v1cs.ROcs.AI

TL;DR

Learning-based control often focuses on action labels, while higher-order temporal consistency remains under-supervised, a concern for stability and robustness in real-world control. The paper adds first-order action supervision to existing policy classes and offline RL methods without structural modifications. It reports broad benchmark gains, including improved scores across OGBench policy paradigms and stronger generalization in small-data settings.

  • Problem

    Learning-based control methods commonly supervise zeroth-order action labels while neglecting higher-order temporal consistency, despite concerns about instability and robustness in real-world deployment.

    Source: §1 · p. 1

  • Method

    The method uses policy time derivatives to add first-order action supervision to deterministic, stochastic, and flow policies, with plug-and-play integration into offline RL.

    Source: §1 · p. 2, §3.2 · p. 5

  • Results

    Across 50 OGBench tasks, first-order supervision yields consistent gains across three policy paradigms, including a ReBRAC score increase from 26 to 61 on antmaze-giant-navigate.

    Source: §1 · p. 2, §5.2 · p. 8

  • Takeaways & Limitations

    The scheme offers a lightweight enhancement for existing offline RL algorithms and supports stable, high performance in challenging small-dataset settings.

    Source: §1 · p. 2, §6 · p. 10

    Pixel-space supervision can be computationally expensive and semantically noisy, while coarse sampling can make finite-difference tangents poor approximations of local dynamics.

    Source: A Limitations · p. 16

Abstract

from arXiv · show

Modern data-driven decision-making methods, such as imitation learning (IL) and reinforcement learning (RL), have achieved great success in solving many complex tasks. However, these methods often suffer from serious control instability and robustness issues when applied in real-world applications such as robotics and autonomous driving, posing notable challenges for their practical deployment. We argue that this instability issue stems largely from their limitations in solely supervising and optimizing zeroth-order actions (i.e., the action labels), failing to account for higher-order action dynamics and temporal consistency. In this paper, we show that simultaneously supervising both zeroth- and first-order actions can dramatically enhance policies' performance and control robustness. To achieve this, we introduce a novel and elegant loss scheme supported by formal theoretical guarantees that can equip any off-the-shelf policy model (e.g., deterministic, stochastic, or flow policies) with the capability for higher-order action supervision, without requiring any structural modifications. Moreover, our proposed method can serve as a lightweight plug-and-play module that seamlessly integrates with a broad spectrum of existing offline RL frameworks. Extensive evaluations on OGBench and D4RL demonstrate that our approach yields substantial performance and robustness improvements across a wide range of continuous control environments. Notably, our method can also enhance policies' out-of-distribution (OOD) generalization capability in the challenging low-data regime, making it an ideal tool in tackling many real-world control problems.

1 Introduction

Learning-based control methods can be unstable and poorly robust because they primarily supervise action labels rather than higher-order temporal consistency. The paper introduces a structural-change-free scheme for supervising both action levels and reports improved performance across policy paradigms and benchmarks.

1 Introduction

  • Learning-based methods can produce nonsmooth, jittering actions and generalize poorly under sparse offline data, complicating real-world deployment.The introduction links high-frequency fluctuations to mechanical wear, energy use, safety risks, and reduced reliability.Source: §1 · pp. 1–2
  • Existing smoothness penalties require task-specific design and tuning, while specialized architectures can restrict expressivity.The paper motivates a general policy-model enhancement as an alternative to these approaches.Source: §1 · p. 2
  • Higher-order supervision equips deterministic, stochastic, and flow policies with simultaneous zeroth- and first-order action supervision without structural modifications.The scheme uses the policy model’s time derivative and is presented as a plug-and-play enhancement.Source: §1 · p. 2
  • The paper reports consistent performance gains across policy paradigms and improved out-of-distribution generalization in small-dataset settings.Its maze2d illustration contrasts noisy first-order actions and a suboptimal BC trajectory with smoother actions and optimal navigation under the proposed supervision.Source: §1 · p. 2, Figure 1 · p. 2

2 Related work

Prior work addresses policy smoothness through penalties or architectural constraints, while offline RL spans deterministic, stochastic, and generative policy classes. The paper identifies higher-order action supervision in offline RL and the use of policy-derived action derivatives in HJB methods as gaps.

2 Related work

  • Penalty-based smoothness methods may require synthetic-state sampling that risks OOD extrapolation in offline learning; architectural constraints can limit expressivity and add inference cost.The cited prior work also includes TaSIL, whose complex evaluation procedure is described as limiting its practicality.Source: §2 · p. 3
  • Offline RL methods use deterministic, stochastic, or generative policies, but the surveyed methods learn only in zeroth-order action spaces.The paper states that higher-order action supervision in offline RL remains unexplored.Source: §2 · p. 3
  • HJB-based continuous-time RL requires first-order action information, but existing methods may approximate it crudely or replace it with a fixed bound.The paper links these substitutions to degraded value estimation and suboptimal performance.Source: §2 · p. 3

3 Extending Standard Policy Classes for Higher-Order Action Supervision

The paper extends standard policy classes to supervise zeroth- and first-order actions using policy derivatives, without changing the policy structure. It gives deterministic, stochastic, and flow-policy instantiations and theoretical error bounds for imitation learning.

3 Extending Standard Policy Classes for Higher-Order Action Supervision

  • The proposed scheme extends standard deterministic, stochastic, and generative flow policies to jointly supervise zeroth- and first-order actions without structural changes.In offline learning, the state and action derivatives are approximated from consecutive dataset samples.Source: §3 · p. 3, §3.1 · p. 4

3.2 Supervising Both Zeroth and First-Order Action for Imitation Learning

  • For deterministic policies, the first-order loss regresses the policy derivative along the state derivative toward the dataset action derivative, alongside the original behavior-cloning loss.The derivative can be evaluated with a Jacobian-vector product, without forming the full policy Jacobian.Source: §3.2 · p. 4
  • For Gaussian policies, the paper derives a first-order action distribution from an Ornstein–Uhlenbeck stochastic process and uses its derivative likelihood for supervision.It distinguishes this stochastic-process formulation from a simpler mean-supervision variant.Source: §3.2 · pp. 4–5
  • For flow policies, the method regresses the physical-time change in the velocity field toward the observed action derivative, complementing standard flow matching.The action-dependent term is omitted because evaluating it increased computational cost without improving performance.Source: §3.2 · p. 5

3.3 Theoretical Analysis

  • Under stated assumptions, Theorem 1 bounds policy error using its initial value and accumulated first-order discrepancy; Theorem 2 bounds closed-loop state deviation using policy mismatch and environment error amplification.The paper says stronger first-order supervision yields tighter bounds on policy error and, consequently, long-term state deviation.Source: §3.3 · p. 6

4 Higher-Order Policy Optimization in Offline RL

The paper incorporates higher-order action information into offline RL through HJB value learning and a first-order policy loss. These additions are designed as plug-and-play modifications to existing algorithms.

4 Higher-Order Policy Optimization in Offline RL

  • The extended policy supplies action derivatives for HJB-based continuous-time RL, where value learning depends on changes along local state-action flow.The paper positions this as a way to use policy-derived action derivatives in critic learning.Source: §4 · p. 6, §4.1 · p. 6

4.1 HJB Equation and Value Learning with First-Order Supervision

  • The HJB value objective adds directional terms for state and action evolution, addressing the lack of temporal-direction alignment in ordinary scalar Bellman regression.The state derivative comes from dataset transitions, while the action derivative is obtained from the learned policy.Source: §4.1 · pp. 6–7

4.2 Higher-Order Policy Learning

  • Offline policy optimization adds a first-order imitation loss to the algorithm’s existing Q-based objective and behavior regularization.The method balances the added loss with a hyperparameter and describes both value- and policy-learning modifications as minimal.Source: §4.2 · p. 7

5 Experiments

The experiments evaluate first-order action supervision across offline RL policy families on OGBench and low-data D4RL, then examine its effects and alternatives. Results report broad performance gains, smoother action evolution, and benefits from both policy and value constraints.

5.1 Experimental Setup

  • The experiments benchmark representative deterministic, stochastic, and flow-matching offline RL policies against first-order-supervised variants on OGBench and 10k-transition D4RL tasks.The setup includes TD3+BC, ReBRAC and IQL variants, and IFQL.Source: §5 · p. 7, §5.1 · pp. 7–8

5.2 Evaluation Results

  • Across 50 OGBench tasks, first-order supervision consistently improves all three policy paradigms; ReBRAC rises from 26 to 61 on antmaze-giant-navigate and from 22 to 55 on humanoidmaze-medium-navigate.Table 1 aggregates scores across 10 task categories, each containing five sub-tasks.Source: Table 1 · p. 8, §5.2 · p. 8
  • On 10k D4RL datasets, IQL rises from 55.7 to 99.5 on hopper-expert-v2, while TD3+BC rises from 2.7 to 82.3 on walker2d-expert-v2.Standard deviations are smaller across most tasks; Table 2 reports results averaged over five seeds.Source: Table 2 · p. 8, §5.2 · p. 9
  • On object-centric play tasks, IFQL improves cube-double-play from 14 to 42 and scene-play from 30 to 56; scene-play-singletask-task2-v0 rises from 0 to 88.The reported relative improvements are 200% for cube-double-play and 87% for scene-play.Source: §5.2 · p. 8
  • With 10k transitions, supervision concentrates rollout action-variation distributions around small values, unlike vanilla BC’s dispersed distribution; this pattern is more pronounced for TD3+BC.The figures aggregate action-variation distributions over 10 trajectories.Source: §5.2 · p. 9, Figure 2 · p. 9, Figure 3 · p. 9

5.3 Ablation experiments

  • Against action-smoothness alternatives, the method improves consistently across tasks; penalty-based methods risk OOD errors on synthetic states, while LipsNet limits expressivity.The comparison uses TD3+BC as the shared backbone and covers 10k D4RL MuJoCo datasets.Source: §5.3 · pp. 9–10, Table 3 · p. 10
  • The auxiliary first-order prediction head brings marginal gains and substantially underperforms the proposed method, while adding outputs and requiring architectural changes.The authors attribute the gap partly to conflicting gradients between zeroth- and first-order objectives.Source: §5.3 · p. 10
  • Sweeping the value-consistency weight λ_Q from 0 to 1 yields steady gains, and increasing the action-supervision weight λ_˙a from 0 to 1 also improves over standard IFQL.Each weight is varied while the other is set to zero; the action-supervision result shows gains without the value regularizer.Source: §5.3 · p. 10

6 Conclusion

The paper concludes that first-order action supervision can improve control smoothness, robustness, and OOD generalization, while fitting existing policy models without structural changes. It also reports an IFQL ablation of value-regularization and first-order action-supervision weights.

6 Conclusion

  • Figure 4 examines IFQL performance while varying value-regularization weight λQ and first-order action-supervision weight λȧ.The caption identifies the two weights but does not state the ablation outcomes.Source: Figure 4 · p. 10
  • The method adds first-order action supervision to off-the-shelf policies without structural modifications, including deterministic, stochastic, and generative flow policies.The paper also reports formal guarantees and integration with the continuous-time HJB framework.Source: §6 · p. 10
  • Evaluations report improvements in control smoothness, robustness, and OOD generalization, particularly in small-dataset settings.The paper describes the method as a promising tool for real-world control problems.Source: §6 · p. 10

A Limitations

The limitations concern raw-pixel supervision and finite-difference estimates of local derivatives. The additional-results section outlines further benchmark, imitation-learning, and ablation results.

A Limitations

  • Raw-pixel first-order supervision can be computationally expensive and semantically noisy; a compact state representation is suggested for task-relevant derivatives.The limitation is the pixel coordinate system, not incompatibility of higher-order supervision with visual inputs.Source: A Limitations · p. 16
  • With coarse sampling, finite-difference tangents may poorly approximate true local dynamics; interpolation or filtering can smooth trajectories before supervision.The limitation concerns the quality of local derivative estimates.Source: A Limitations · p. 16

B Additional Results

  • The additional-results section reports per-task OGBench results, full-dataset D4RL scores, imitation-learning results, and further ablations.The passage lists these as the section’s contents.Source: B Additional Results · p. 16

B.1 Additional Evaluation Results

Across OGBench tasks and D4RL settings, first-order action supervision improves results across policy paradigms and can lift some zero-score baselines to substantial scores. Additional evaluations report smoother action evolution, modest training-time overhead, and faster online-RL convergence.

B.1 Additional Evaluation Results

  • Across diverse offline policy paradigms, supervision improves performance and can raise IFQL on scene-play-singletask-task2-v0 from 0 to 88.The OGBench evaluation covers 50 tasks; reported examples also include IFQL gains from 9 to 40 and ReBRAC gains from 49 to 96.Source: §B.1 · pp. 16–17, Table 4 · p. 16
  • On full-dataset D4RL MuJoCo locomotion tasks, supervision yields consistent performance improvements across all four base algorithms.The evaluation is reported in Table 5.Source: §B.1 · p. 17, Table 5 · p. 17
  • On deterministic Walker2d-e BC, adding first-order supervision raises the score from 26.7 to 91.1, an absolute improvement of 64.4.The passage attributes the gain to addressing compounding rollout errors through temporal regularization.Source: §B.1 · pp. 17–18, Table 6 · p. 17
  • Across additional imitation-learning and offline-RL cases, supervised variants show smoother learning curves and more concentrated action-variation distributions.The rollout-distribution comparison aggregates ||ȧ|| over 10 trajectories; learning results are averaged over five seeds.Source: §B.1 · p. 18, Figure 5 · p. 18, Figure 6 · p. 18
  • Adding first-order supervision only moderately increases training time across four policy parameterizations.The comparison covers hopper-medium-v2 and cube-double; runs use the same number of training steps on a single GPU.Source: §B.1 · p. 18, Table 7 · p. 18
  • In online SAC evaluations, first-order supervision yields faster convergence, especially at 300k steps, and reduced performance variance across seeds.The evaluation uses three MuJoCo benchmarks over 1M training steps.Source: §B.1 · pp. 18–19, Table 8 · p. 19

B.2 Additional Ablation Results

Additional ablations find benefits from both policy-side and value-side regularization and compare the proposed state-directional approximation with an action-dependent alternative. The appendix also identifies derivations for stochastic Gaussian and flow-based policies.

B.2 Additional Ablation Results

  • In TD3+BC and IQL on 10k D4RL expert tasks, increasing either λȧ or λQ from 0 to 1 yields steady performance gains.The weights are swept independently, with the other set to zero; results average five seeds.Source: §B.2 · p. 19
  • AK adds an action-dependent derivative term to the state-directional term and uses the same objective as Eq. (7).On the linear interpolation path, ȧk = kȧ1.Source: §B.2 · p. 19
  • Across tested OGBench environments, Ours matches or improves on IFQL for every task and generally outperforms AK, whose performance drops on some tasks.AK’s drops are especially notable on cube-double and puzzle-4x4.Source: §B.2 · p. 20, Table 9 · p. 20
  • Compared with Ours, AK nearly doubles computation per update (+99.3%) and uses 11.0% more GPU memory.These results support the empirical choice of the state-directional approximation.Source: §B.2 · p. 20, Table 10 · p. 20

C Derivation of the Higher-Order Supervision

C.1 First-Order Stochastic Gaussian Policies Derivation in Eq. (3)

The appendices derive first-order action supervision for stochastic Gaussian and flow policies, then develop assumptions and proofs for policy-error bounds. The analysis uses differentiability and regularity conditions to relate policy error along trajectories to state deviation.

C.1 First-Order Stochastic Gaussian Policies Derivation in Eq. (3)

  • A stationary OU noise process yields stochastic actions, whose time derivative induces a first-order policy distribution approximated in discrete time by a unit-step finite difference.The action uses element-wise noise scaling; the derivation applies Itô’s formula to obtain the action-process dynamics.Source: §C.1 · pp. 20–21

C.2 First-Order Flow Policies Derivation in Eq. (6)

  • For flow policies, differentiating the action-generation path provides a sample-wise target for the first-order field, which is supervised in a flow-matching manner.The implementation approximates the physical-time field derivative with its state derivative along the state velocity at sampled flow times.Source: §C.2 · p. 22

D Theoretical Analysis

  • The theoretical-analysis section states that it establishes notation and assumptions before proving Theorems 1 and 2.This section overview identifies the proof structure but does not state the theorem results.Source: D Theoretical Analysis · p. 22

D.1 Notation

  • The analysis defines learned and expert policies, their trajectories, policy error, and state deviation to formalize the comparison.It also defines the segment connecting the learned and expert states at each time.Source: §D.1 · p. 22

D.2 Assumptions

  • The assumptions require a compact convex state domain, differentiable policies, Lipschitz dynamics and expert policy, and bounded policy-error Jacobian.The dynamics are assumed Lipschitz in both state and action.Source: §D.2 · pp. 22–23

D.3 Auxiliary Lemmas

  • Lemma 1 uses a line segment between two states and integrates the policy-error derivative, then applies that relation to learned and expert states.The proof relies on differentiability over the convex domain.Source: §D.3 · p. 23

D.4 Proof of Theorem 1

  • Theorem 1’s proof differentiates the policy error along the expert trajectory and integrates its time derivative.The analysis first introduces the expert action derivative along that trajectory.Source: §D.4 · p. 24

D.5 Proof of Theorem 2

Theorem 2 bounds rollout state deviation under stated regularity assumptions and matching initial states. The evaluation uses offline benchmark datasets without environment interaction.

D.5 Proof of Theorem 2

  • Under Assumptions 1–4 and δ(0) = 0, Theorem 2 bounds state deviation using the maximum policy error along the expert trajectory and a constant β.β is defined from the state- and action-Lipschitz constants, the expert-policy Lipschitz constant, and the policy-error Jacobian bound.Source: §D.5 · p. 24
  • The proof bounds the rate of state deviation using policy error and trajectory deviation, then applies Gronwall’s lemma.The derivation combines Lipschitz bounds for dynamics and the expert policy with the bounded policy-error Jacobian.Source: §D.5 · p. 25

E Experimental Details

  • The evaluation trains and tests methods on offline datasets from OGBench and D4RL, without online environment interaction.These benchmarks cover diverse task structures, state and action dimensionalities, and dataset qualities.Source: E Experimental Details · p. 25

E.1 Benchmarks and Tasks

The evaluation spans 50 OGBench tasks across 10 environments and D4RL MuJoCo locomotion datasets. It also includes a trajectory-sampled 10k-transition setting for studying data efficiency.

OGBench

  • OGBench evaluation covers 50 reward-based single-task state-based tasks across 10 environments, including locomotion, object interaction, and robotic manipulation.Scene and puzzle tasks add long-horizon multi-object manipulation and combinatorial reasoning challenges, respectively.Source: §E.1 · p. 26
  • OGBench tasks use state dimensions from 28 to 83 and action dimensions from 5 to 21, with standard datasets aggregating trajectories from diverse, often suboptimal behaviors.The heterogeneous data makes extracting task-relevant behavior challenging for offline RL.Source: §E.1 · p. 26

D4RL

  • D4RL evaluation uses halfcheetah, hopper, and walker2d with medium, medium-replay, medium-expert, and expert datasets, plus a 10k-transition small-data setting.The small setting samples complete trajectories to preserve temporal continuity for first-order supervision based on consecutive transitions.Source: §E.1 · p. 26

E.2 Methods and Hyperparameters

This appendix documents the application of first-order action supervision across four offline RL algorithms, along with its potential impact, limitations, and reproducibility information, and reports no important, original, or non-standard LLM components in the core method development.

E.2 Methods and Hyperparameters

  • The method is integrated into TD3+BC, ReBRAC, IQL, and IFQL on a shared FQL codebase, keeping baseline settings while adding value- and policy-side supervision weights.TD3+BC sweeps its Q-value scaling coefficient and uses Q-network layer normalization on challenging OGBench tasks; ReBRAC, IQL, and IFQL use algorithm-specific settings and supervision weights across OGBench and D4RL.Source: §E.2 · pp. 26–27

F Broader Impact

  • The authors present the scheme as a lightweight, structurally unchanged way to improve smoothness, stability, and OOD generalization, while noting that policies may inherit unsafe or suboptimal dataset behaviors.They identify robotics, drone control, and autonomous driving as safety-critical application domains.Source: F Broader Impact · p. 27

2. Limitations

  • The paper says it discusses limitations in Appendix A and provides theoretical assumptions and complete proofs in the paper and appendices.It also reports computational training time and hardware resources in Table 7.Source: §2 · pp. 28, 29, 31

NeurIPS Paper Checklist

  • The checklist reports that code is withheld for anonymity but planned for GitHub release upon acceptance, and that training details and experiment settings are specified in the appendices.It also states that table captions clarify how results and standard deviations were calculated.Source: §2 · p. 30

E.2 Methods and Hyperparameters

  • The core method development does not involve LLMs as important, original, or non-standard components.The response directs readers to the NeurIPS handbook for guidance on what should be described under the LLM policy.Source: §2 · p. 34
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